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Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> 5 = 6 - 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 1 = 2 - 1
([],2)
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 100 => 2 = 3 - 1
([],3)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 1000 => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> 10110 => 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 10110 => 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 10000 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 11010 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 11010 => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> 11010 => 5 = 6 - 1
Description
The number of inversions of a binary word.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00125: Posets dual posetPosets
Mp00195: Posets order idealsLattices
St001622: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],2)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([],3)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> 4 = 5 - 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> 5 = 6 - 1
Description
The number of join-irreducible elements of a lattice. An element j of a lattice L is '''join irreducible''' if it is not the least element and if j=xy, then j{x,y} for all x,yL.
Matching statistic: St000147
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 4
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 4
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> 5
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> [5,3,2]
=> 5
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> [5,2,2]
=> 5
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> [5,3,2]
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> [5,2,2]
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> [6,2,2]
=> 6
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> 6
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> [6,2]
=> 6
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> 6
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> [6,2,2]
=> 6
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> [6,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> [6,2,2]
=> 6
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> [6,4]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> 6
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> 6
Description
The largest part of an integer partition.
Matching statistic: St000384
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 4
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 4
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> 5
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> [5,3,2]
=> 5
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> [5,2,2]
=> 5
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> [5,3,2]
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> [5,2,2]
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> [6,2,2]
=> 6
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> 6
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> [6,2]
=> 6
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> 6
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> [6,2,2]
=> 6
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> [6,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> [6,2,2]
=> 6
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> [6,4]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> 6
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> 6
Description
The maximal part of the shifted composition of an integer partition. A partition λ=(λ1,,λk) is shifted into a composition by adding i1 to the i-th part. The statistic is then maxi{λi+i1}. See also [[St000380]].
Matching statistic: St000784
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 4
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 4
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> 5
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> [5,3,2]
=> 5
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> [5,2,2]
=> 5
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> [5,3,2]
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> [5,2,2]
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> [6,2,2]
=> 6
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> 6
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> [6,2]
=> 6
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> 6
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> [6,2,2]
=> 6
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> [6,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> [6,2,2]
=> 6
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> [6,4]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> 6
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> 6
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St000018
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5 = 6 - 1
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions (i,i+1) needed to write π. Thus, it is also the Coxeter length of π.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000246: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
Description
The number of non-inversions of a permutation. For a permutation of {1,,n}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
Matching statistic: St000290
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00316: Binary words inverse Foata bijectionBinary words
St000290: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 10 => 1 = 2 - 1
([],2)
=> [1,1]
=> 110 => 110 => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 100 => 010 => 2 = 3 - 1
([],3)
=> [1,1,1]
=> 1110 => 1110 => 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 1010 => 0110 => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1010 => 0110 => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 1000 => 0010 => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 1010 => 0110 => 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01110 => 4 = 5 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> 10110 => 01110 => 4 = 5 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 10110 => 01110 => 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 01110 => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 01110 => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 1010 => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1010 => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1010 => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 10000 => 00010 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 00110 => 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 100010 => 000110 => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 11010 => 10110 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> 11010 => 10110 => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 100010 => 000110 => 5 = 6 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> 10100 => 10010 => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 100010 => 000110 => 5 = 6 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 100010 => 000110 => 5 = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 001110 => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> 11010 => 10110 => 5 = 6 - 1
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length n with a zeros, the generating function for the major index is the q-binomial coefficient \binom{n}{a}_q.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000395The sum of the heights of the peaks of a Dyck path. St001746The coalition number of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001759The Rajchgot index of a permutation. St000189The number of elements in the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001342The number of vertices in the center of a graph. St001725The harmonious chromatic number of a graph. St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000144The pyramid weight of the Dyck path. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001120The length of a longest path in a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000719The number of alignments in a perfect matching. St001645The pebbling number of a connected graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000093The cardinality of a maximal independent set of vertices of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001875The number of simple modules with projective dimension at most 1. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset.